Optimal. Leaf size=126 \[ -\frac {\text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {x^2 \sinh ^2(a+b x)}{2 b}+\frac {x^2}{4 b}-\frac {x^3}{3} \]
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Rubi [A] time = 0.20, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 9, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {5450, 5372, 3310, 30, 3716, 2190, 2531, 2282, 6589} \[ \frac {x \text {PolyLog}\left (2,e^{2 (a+b x)}\right )}{b^2}-\frac {\text {PolyLog}\left (3,e^{2 (a+b x)}\right )}{2 b^3}+\frac {\sinh ^2(a+b x)}{4 b^3}-\frac {x \sinh (a+b x) \cosh (a+b x)}{2 b^2}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {x^2 \sinh ^2(a+b x)}{2 b}+\frac {x^2}{4 b}-\frac {x^3}{3} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2190
Rule 2282
Rule 2531
Rule 3310
Rule 3716
Rule 5372
Rule 5450
Rule 6589
Rubi steps
\begin {align*} \int x^2 \cosh ^2(a+b x) \coth (a+b x) \, dx &=\int x^2 \coth (a+b x) \, dx+\int x^2 \cosh (a+b x) \sinh (a+b x) \, dx\\ &=-\frac {x^3}{3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}-2 \int \frac {e^{2 (a+b x)} x^2}{1-e^{2 (a+b x)}} \, dx-\frac {\int x \sinh ^2(a+b x) \, dx}{b}\\ &=-\frac {x^3}{3}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}+\frac {\int x \, dx}{2 b}-\frac {2 \int x \log \left (1-e^{2 (a+b x)}\right ) \, dx}{b}\\ &=\frac {x^2}{4 b}-\frac {x^3}{3}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}-\frac {\int \text {Li}_2\left (e^{2 (a+b x)}\right ) \, dx}{b^2}\\ &=\frac {x^2}{4 b}-\frac {x^3}{3}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}-\frac {\operatorname {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 (a+b x)}\right )}{2 b^3}\\ &=\frac {x^2}{4 b}-\frac {x^3}{3}+\frac {x^2 \log \left (1-e^{2 (a+b x)}\right )}{b}+\frac {x \text {Li}_2\left (e^{2 (a+b x)}\right )}{b^2}-\frac {\text {Li}_3\left (e^{2 (a+b x)}\right )}{2 b^3}-\frac {x \cosh (a+b x) \sinh (a+b x)}{2 b^2}+\frac {\sinh ^2(a+b x)}{4 b^3}+\frac {x^2 \sinh ^2(a+b x)}{2 b}\\ \end {align*}
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Mathematica [A] time = 2.59, size = 178, normalized size = 1.41 \[ \frac {\sinh (a) (\sinh (a)+\cosh (a)) \left (24 b^2 x^2 \log \left (1-e^{-a-b x}\right )+24 b^2 x^2 \log \left (e^{-a-b x}+1\right )+6 b^2 x^2 \cosh (2 (a+b x))-48 b x \text {Li}_2\left (-e^{-a-b x}\right )-48 b x \text {Li}_2\left (e^{-a-b x}\right )-48 \text {Li}_3\left (-e^{-a-b x}\right )-48 \text {Li}_3\left (e^{-a-b x}\right )-6 b x \sinh (2 (a+b x))+3 \cosh (2 (a+b x))+8 b^3 x^3\right )}{12 \left (e^{2 a}-1\right ) b^3} \]
Antiderivative was successfully verified.
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fricas [C] time = 0.92, size = 697, normalized size = 5.53 \[ \frac {3 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} \cosh \left (b x + a\right )^{4} + 12 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + 3 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} \sinh \left (b x + a\right )^{4} + 6 \, b^{2} x^{2} - 16 \, {\left (b^{3} x^{3} + 2 \, a^{3}\right )} \cosh \left (b x + a\right )^{2} - 2 \, {\left (8 \, b^{3} x^{3} + 16 \, a^{3} - 9 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{2} + 6 \, b x + 96 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2}\right )} {\rm Li}_2\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) + 96 \, {\left (b x \cosh \left (b x + a\right )^{2} + 2 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b x \sinh \left (b x + a\right )^{2}\right )} {\rm Li}_2\left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 48 \, {\left (b^{2} x^{2} \cosh \left (b x + a\right )^{2} + 2 \, b^{2} x^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{2} x^{2} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 48 \, {\left (a^{2} \cosh \left (b x + a\right )^{2} + 2 \, a^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + a^{2} \sinh \left (b x + a\right )^{2}\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 48 \, {\left ({\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right )^{2} + 2 \, {\left (b^{2} x^{2} - a^{2}\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + {\left (b^{2} x^{2} - a^{2}\right )} \sinh \left (b x + a\right )^{2}\right )} \log \left (-\cosh \left (b x + a\right ) - \sinh \left (b x + a\right ) + 1\right ) - 96 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )} {\rm polylog}\left (3, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right ) - 96 \, {\left (\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2}\right )} {\rm polylog}\left (3, -\cosh \left (b x + a\right ) - \sinh \left (b x + a\right )\right ) + 4 \, {\left (3 \, {\left (2 \, b^{2} x^{2} - 2 \, b x + 1\right )} \cosh \left (b x + a\right )^{3} - 8 \, {\left (b^{3} x^{3} + 2 \, a^{3}\right )} \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 3}{48 \, {\left (b^{3} \cosh \left (b x + a\right )^{2} + 2 \, b^{3} \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + b^{3} \sinh \left (b x + a\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{2} \cosh \left (b x + a\right )^{3} \operatorname {csch}\left (b x + a\right )\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.62, size = 222, normalized size = 1.76 \[ -\frac {x^{3}}{3}+\frac {\left (2 x^{2} b^{2}-2 b x +1\right ) {\mathrm e}^{2 b x +2 a}}{16 b^{3}}+\frac {\left (2 x^{2} b^{2}+2 b x +1\right ) {\mathrm e}^{-2 b x -2 a}}{16 b^{3}}+\frac {a^{2} \ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{3}}-\frac {2 a^{2} \ln \left ({\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {2 a^{2} x}{b^{2}}+\frac {4 a^{3}}{3 b^{3}}+\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) x^{2}}{b}-\frac {\ln \left (1-{\mathrm e}^{b x +a}\right ) a^{2}}{b^{3}}+\frac {2 \polylog \left (2, {\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 \polylog \left (3, {\mathrm e}^{b x +a}\right )}{b^{3}}+\frac {\ln \left (1+{\mathrm e}^{b x +a}\right ) x^{2}}{b}+\frac {2 \polylog \left (2, -{\mathrm e}^{b x +a}\right ) x}{b^{2}}-\frac {2 \polylog \left (3, -{\mathrm e}^{b x +a}\right )}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.45, size = 171, normalized size = 1.36 \[ -\frac {2}{3} \, x^{3} + \frac {{\left (16 \, b^{3} x^{3} e^{\left (2 \, a\right )} + 3 \, {\left (2 \, b^{2} x^{2} e^{\left (4 \, a\right )} - 2 \, b x e^{\left (4 \, a\right )} + e^{\left (4 \, a\right )}\right )} e^{\left (2 \, b x\right )} + 3 \, {\left (2 \, b^{2} x^{2} + 2 \, b x + 1\right )} e^{\left (-2 \, b x\right )}\right )} e^{\left (-2 \, a\right )}}{48 \, b^{3}} + \frac {b^{2} x^{2} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (-e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(-e^{\left (b x + a\right )})}{b^{3}} + \frac {b^{2} x^{2} \log \left (-e^{\left (b x + a\right )} + 1\right ) + 2 \, b x {\rm Li}_2\left (e^{\left (b x + a\right )}\right ) - 2 \, {\rm Li}_{3}(e^{\left (b x + a\right )})}{b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x^2\,{\mathrm {cosh}\left (a+b\,x\right )}^3}{\mathrm {sinh}\left (a+b\,x\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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